| L(s) = 1 | − 2·2-s + 2·4-s + 2·5-s − 10·7-s − 4·8-s − 3·9-s − 4·10-s + 4·11-s + 12·13-s + 20·14-s + 8·16-s − 12·17-s + 6·18-s + 12·19-s + 4·20-s − 8·22-s − 6·23-s + 5·25-s − 24·26-s − 20·28-s − 18·29-s + 6·31-s − 8·32-s + 24·34-s − 20·35-s − 6·36-s − 12·37-s + ⋯ |
| L(s) = 1 | − 1.41·2-s + 4-s + 0.894·5-s − 3.77·7-s − 1.41·8-s − 9-s − 1.26·10-s + 1.20·11-s + 3.32·13-s + 5.34·14-s + 2·16-s − 2.91·17-s + 1.41·18-s + 2.75·19-s + 0.894·20-s − 1.70·22-s − 1.25·23-s + 25-s − 4.70·26-s − 3.77·28-s − 3.34·29-s + 1.07·31-s − 1.41·32-s + 4.11·34-s − 3.38·35-s − 36-s − 1.97·37-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 3^{8} \cdot 5^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 3^{8} \cdot 5^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.4847763528\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.4847763528\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 2 | $C_2^2$ | \( 1 + p T + p T^{2} + p^{2} T^{3} + p^{2} T^{4} \) |
| 3 | $C_2^2$ | \( 1 + p T^{2} + p^{2} T^{4} \) |
| 5 | $C_2^2$ | \( 1 - 2 T - T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| good | 7 | $C_2$$\times$$C_2^2$ | \( ( 1 + 5 T + p T^{2} )^{2}( 1 + 2 T^{2} + p^{2} T^{4} ) \) |
| 11 | $C_2^2$ | \( ( 1 - 2 T - 7 T^{2} - 2 p T^{3} + p^{2} T^{4} )^{2} \) |
| 13 | $D_4\times C_2$ | \( 1 - 12 T + 36 T^{2} + 12 p T^{3} - 1273 T^{4} + 12 p^{2} T^{5} + 36 p^{2} T^{6} - 12 p^{3} T^{7} + p^{4} T^{8} \) |
| 17 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2}( 1 + 8 T + p T^{2} )^{2} \) |
| 19 | $D_{4}$ | \( ( 1 - 6 T + 44 T^{2} - 6 p T^{3} + p^{2} T^{4} )^{2} \) |
| 23 | $D_4\times C_2$ | \( 1 + 6 T + 9 T^{2} - 6 p T^{3} - 1072 T^{4} - 6 p^{2} T^{5} + 9 p^{2} T^{6} + 6 p^{3} T^{7} + p^{4} T^{8} \) |
| 29 | $D_4\times C_2$ | \( 1 + 18 T + 189 T^{2} + 1458 T^{3} + 8852 T^{4} + 1458 p T^{5} + 189 p^{2} T^{6} + 18 p^{3} T^{7} + p^{4} T^{8} \) |
| 31 | $D_4\times C_2$ | \( 1 - 6 T - 8 T^{2} + 108 T^{3} - 141 T^{4} + 108 p T^{5} - 8 p^{2} T^{6} - 6 p^{3} T^{7} + p^{4} T^{8} \) |
| 37 | $D_4\times C_2$ | \( 1 + 12 T + 72 T^{2} + 372 T^{3} + 1886 T^{4} + 372 p T^{5} + 72 p^{2} T^{6} + 12 p^{3} T^{7} + p^{4} T^{8} \) |
| 41 | $D_4\times C_2$ | \( 1 + 12 T + 133 T^{2} + 1020 T^{3} + 7512 T^{4} + 1020 p T^{5} + 133 p^{2} T^{6} + 12 p^{3} T^{7} + p^{4} T^{8} \) |
| 43 | $D_4\times C_2$ | \( 1 - 18 T + 90 T^{2} + 480 T^{3} - 7489 T^{4} + 480 p T^{5} + 90 p^{2} T^{6} - 18 p^{3} T^{7} + p^{4} T^{8} \) |
| 47 | $D_4\times C_2$ | \( 1 + 12 T + 45 T^{2} - 228 T^{3} - 3820 T^{4} - 228 p T^{5} + 45 p^{2} T^{6} + 12 p^{3} T^{7} + p^{4} T^{8} \) |
| 53 | $C_2^3$ | \( 1 - 2914 T^{4} + p^{4} T^{8} \) |
| 59 | $D_4\times C_2$ | \( 1 - 18 T + 204 T^{2} - 1728 T^{3} + 12107 T^{4} - 1728 p T^{5} + 204 p^{2} T^{6} - 18 p^{3} T^{7} + p^{4} T^{8} \) |
| 61 | $C_2$$\times$$C_2^2$ | \( ( 1 - 10 T + p T^{2} )^{2}( 1 - 10 T + 39 T^{2} - 10 p T^{3} + p^{2} T^{4} ) \) |
| 67 | $D_4\times C_2$ | \( 1 - 12 T + 45 T^{2} + 972 T^{3} - 11812 T^{4} + 972 p T^{5} + 45 p^{2} T^{6} - 12 p^{3} T^{7} + p^{4} T^{8} \) |
| 71 | $C_2^2$ | \( ( 1 + 2 T^{2} + p^{2} T^{4} )^{2} \) |
| 73 | $D_4\times C_2$ | \( 1 + 20 T + 200 T^{2} + 1380 T^{3} + 9506 T^{4} + 1380 p T^{5} + 200 p^{2} T^{6} + 20 p^{3} T^{7} + p^{4} T^{8} \) |
| 79 | $D_4\times C_2$ | \( 1 + 18 T + 284 T^{2} + 3168 T^{3} + 33267 T^{4} + 3168 p T^{5} + 284 p^{2} T^{6} + 18 p^{3} T^{7} + p^{4} T^{8} \) |
| 83 | $D_4\times C_2$ | \( 1 - 26 T + 365 T^{2} - 3554 T^{3} + 32320 T^{4} - 3554 p T^{5} + 365 p^{2} T^{6} - 26 p^{3} T^{7} + p^{4} T^{8} \) |
| 89 | $D_{4}$ | \( ( 1 - 12 T + 139 T^{2} - 12 p T^{3} + p^{2} T^{4} )^{2} \) |
| 97 | $C_2^2$$\times$$C_2^2$ | \( ( 1 - 8 T - 33 T^{2} - 8 p T^{3} + p^{2} T^{4} )( 1 + 18 T + 227 T^{2} + 18 p T^{3} + p^{2} T^{4} ) \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.460760084699498731497810341785, −8.376822729457030897708478590697, −8.137029556860488347305313635560, −7.30644559387283716890224394293, −7.26842937255263520728776205640, −6.76617208939441047528234753472, −6.68123091850912379968081682033, −6.66922759777581424519476019324, −6.52743522837608655609495795542, −5.99253964535360062305549473134, −5.78363202728331103823639317830, −5.69062507840100974981861511378, −5.52293360976887690021326651482, −5.31956495790572260421940682472, −4.18092284856304475780353251093, −4.10959225765517939633917953384, −3.75227504656567874975225134702, −3.40554441284297033781978645065, −3.37385539124051500239328264006, −3.09706688560279812727649510890, −2.63599386515490436787390014667, −2.06699578117492395839295005303, −1.78185639619690260800517351482, −0.870115731589868472600962408350, −0.48351030286172244255775552869,
0.48351030286172244255775552869, 0.870115731589868472600962408350, 1.78185639619690260800517351482, 2.06699578117492395839295005303, 2.63599386515490436787390014667, 3.09706688560279812727649510890, 3.37385539124051500239328264006, 3.40554441284297033781978645065, 3.75227504656567874975225134702, 4.10959225765517939633917953384, 4.18092284856304475780353251093, 5.31956495790572260421940682472, 5.52293360976887690021326651482, 5.69062507840100974981861511378, 5.78363202728331103823639317830, 5.99253964535360062305549473134, 6.52743522837608655609495795542, 6.66922759777581424519476019324, 6.68123091850912379968081682033, 6.76617208939441047528234753472, 7.26842937255263520728776205640, 7.30644559387283716890224394293, 8.137029556860488347305313635560, 8.376822729457030897708478590697, 8.460760084699498731497810341785
